Question

Consider a monoatomic ideal gas undergoing the following cycle: starting point (a), pressure increases at a constant volume reaching point (b), then the gas expands adiabatically until pressure reaches the initial value (point c), and then the gas is compressed at a constant pressure until the volume reaches the initial value back to point (a). The amount of gas is 1 mole. Monoatomic gas means it has only 3 degrees of freedom and the adiabatic constant gamma is 5/3.

Sketch the cycle at the PV-diagram 1.1 ANSWER

1.2 [2pt] Sketch the cycle along with the two isotherms which cross the points of maximal pressure and maximal volume, respectively.

1.3 [2pt] Indicate in the PV-diagram the points of lowest and highest temperatures during the cycle.

Suppose the lowest pressure in the cycle is P1, the lowest volume is V1, and the highest volume is V2. Let us also label the highest temperature during the cycle as TH and the lowest temperature during the cycle as TC.

1.4 [2pt] Find the highest pressure during the cycle. Express the answer in terms of P1, V1, and V2.

1.5 [2pt] Find the gas temperatures at points (a), (b), and (c) and express the answers in terms of P1, V1, and V2.

1.6 [2pt] Suppose we only know the ratio V1/V2 and don’t know P1. Calculate the ratio of TH to TC.

1.7 [2pt] Express the temperature at point c (the end of the adiabatic process) via the temperatures TH and TC only.

1.8 [1pt] What is the internal energy change in the adiabatic process (b->c) in terms of P1, V1, V2?

1.9 [1pt] What is the internal energy change in the adiabatic process (b->c) in terms of TC and TH?

1.10 [1pt] What is the internal energy change in the constant volume process (a->b) in terms of P1, V1, V2?

1.11 [1pt] What is the internal energy change in the constant volume process (a->b) in terms of TH and TC?

1.12 [1pt] What is the internal energy change in the constant pressure process (c->a) in terms of P1, V1, V2?

1.13 [1pt] What is the internal energy change in the constant pressure process (c->a) in terms of TH and TC?

1.12 [1pt] What is the total internal energy change during the cycle?

1.13 [1pt] Does the gas produce positive or negative work in the cycle?

1.14 [1p] Calculate the work done by the gas during the adiabatic process (b->c). Express the answer in terms of P1, V1, V2.

1.15 [1p] Calculate the work done by the gas during the adiabatic process (b->c). Express the answer using TC and TH.

1.16 [2pt] Calculate the work done by the gas during the cycle. Express it in terms of TH and TC

1.17 [2pt] Indicate at the cycle in the PV diagram during which steps the gas receives/releases heat

1.18 [1pt] Does the gas overall gain or loose heat during the cycle?

1.19 [1pt] Does the cycle correspond to a heat engine or to a refrigerator?

1.20 [2pt] Calculate the amounts of heat received and lost by the gas during the cycle. Express your answers in terms of TH and TC.
1.21 [2pt] Calculate the efficiency of our cycle and express it via the ratio TC/TH 1.21 ANSWER

1.22 [2pt] Can your answer for the efficiency in Q1.21 be larger than 1? Hint: denote TC/TH = x, define efficiency as a function of x <1. You can either plot the function using your favorite computer program or study its properties using calculus.

1.23 [2pt] At the same PV diagram, sketch our cycle along with a Carnot cycle which could be constructed using the two isotherms corresponding to the temperatures TH and TC

1.24 [1pt] What is the efficiency of the Carnot cycle from Q1.23 in terms of TH and TC

1.25 [2p] Compare the efficiency obtained from Q1.21 and from 1.24 at all possible values of TC/TH. Which of the two cycles is more efficient in converting heat to work? Show your proof.

Answer 1